Final answer:
To find dy/dx at the point (1,1), differentiate the given equation implicitly with respect to x. Substitute the coordinates of the point (1,1) into the equation and solve for dy/dx. The value of dy/dx at the point (1,1) is 1/2.
Step-by-step explanation:
To find dy/dx at the point (1,1), we can differentiate the given equation implicitly with respect to x.
First, we'll differentiate each term:
- For the term x, the derivative is 1, since the derivative of x with respect to x is 1.
- For the term 2xy, we'll use the product rule: Differentiating x gives 1, differentiating y gives dy/dx, and multiplying by 2 gives 2(1)(dy/dx) = 2(dy/dx).
- For the term -y², we'll use the chain rule: Differentiating y² gives 2y, and multiplying by -1 gives -2y.
- For the constant term 2, the derivative is 0, since the derivative of a constant is 0.
Putting it all together, we have the equation 1 + 2(dy/dx)y - 2y = 0. Substituting the point (1,1), we get 1 + 2(dy/dx)(1) - 2(1) = 0. Simplifying this equation gives 1 + 2(dy/dx) - 2 = 0. Combining like terms, we have 2(dy/dx) - 1 = 0. Solving for dy/dx, we get dy/dx = 1/2.